Singlet-triplet filtering and entanglement in a quantum dot structure

PHYSICAL REVIEW B 75, 085302 2007Singlet-triplet filtering and entanglement in a quantum dot structureG. Giavaras, 1,2 J. H. Jefferson, 2 M. Fearn, 2 and C. J. Lambert 11 Department of Physics, Lancaster University, Lancaster LA14YB, England2 QinetiQ, St.Andrews Road, Malvern WR143PS, EnglandReceived 25 October 2006; revised manuscript received 30 November 2006; published 1 February 2007We consider two interacting electrons in a semiconductor quantum dot structure, which consists of a smalldot within a larger dot, and demonstrate a singlet-triplet filtering mechanism which involves spin-dependentresonances and can generate entanglement. By studying the exact time evolution of singlet and triplet states weshow how the degree of both filtering and spin entanglement can be tuned using a time-dependent gate voltage.DOI: 10.1103/PhysRevB.75.085302PACS numbers: 73.21.La, 73.63.b, 73.23.HkIn semiconductor quantum dots which can be formed forexample at the interface of a gated GaAs/AlGaAs heterostructure,we can trap electrons and control their numberprecisely down to 1. 1–4 The fact that the electrons in the dotsare well confined within a region of some nanometers, andthey are therefore isolated from the remaining environmentof the host material, offers not only the opportunity to exploreatomiclike effects and electron correlations 2,3,5,6 in thesolid state, but also to consider the quantum dots as promisingcandidates for nanoelectronic applications. For instance,the proposal for realization of spin and charge qubits indouble quantum dots 7,8 has attracted a lot of interest boththeoretically 9–13 and experimentally 14–17 and the demonstrationof entanglement, which is a necessary ingredient fortwo-qubit gates, is one of the main goals. Many systemswhich include quantum dots make use of the Loss andDiVincenzo 7 exchange-energy mechanism based on the Coulombinteraction and Pauli exclusion to generate spinentanglement.For example, Burkard et al. 18 proposed theuse of quantum dot structures as efficient entangler devicesin solid state. In addition, spin filtering was proposed inquantum dots 19 and spin entanglement creation via Coulombscattering was studied in Refs. 20 and 21. Furthermore, flyingqubits in surface acoustic wave based quantum dots 22–25and static-flying qubits in quantum wires 25,26 have been suggestedand studied as means of entanglement generation betweenelectron spins. The singlet-triplet qubit in a doublequantum dot is also a promising candidate for quantum computation.The two-electron singlet-triplet S z =0 states constitutethe two-level quantum system and as has been shownthis scheme, which is also under experimentalinvestigation, 17 is efficient for universal quantum gates. 27,28In this paper we propose and investigate a semiconductordot structure which can induce a mechanism to filter singlettripletstates and thus to generate maximal spin entanglementbetween two electrons. The operation is controlled by a timedependentgate voltage which can tune the degree of filteringand entanglement.To be specific, we choose material parameters for GaAsand consider the quasi-one-dimensional 1D quantum dotstructure which is shown in Fig. 1a, loaded with two electrons.It consists of a small open dot which is formed in thecenter of a much larger dot. The small dot is such that it canbind only one electron and this means that the Coulombinteraction forces the second electron to occupy the region ofthe large dot. The latter has width L=800 nm80a B * , wherea B * is the effective Bohr radius for GaAs. In this regime thetwo electrons are in the strong correlation regime, 29,30 i.e.,the Coulomb interaction dominates over the kinetic energyand in the ground state the two electrons are localized inregions for which the electrostatic repulsion is minimized.Note that this is the case for both singlet and triplet stateswhich give virtually the same electron distribution and theyare separated with a very small antiferromagnetic exchangeenergy J.To explicitly demonstrate and analyze these effects wehave studied the two-electron problem with exactdiagonalization 31 considering the parabolic-band HamiltonianH = i=1,2− 2 22m * 2x + V dx ii + V c x 1 ,x 2 ,where m * =0.067m o is the effective mass of the electrons forGaAs. We have modelled the quantum dot with theGaussian confining potential V d x=−V o exp−x 2 /2l o 2 , for−400 nm x400 nm and V d x= otherwise. The parametersV o , l o determine the depth and width of the small dot,respectively, and they are chosen to give only a single boundenergy level −1.5 meV. A realistic well depth for the largedot would be a few tens of meV and since for the chosenwidth of dot the energy scale of the relevant singlet andtriplet states is a few meV, the infinite well approximationhas only a small quantitative effect on the results. In practice,the hard wall would be replaced by finite barriers but, again,these may be chosen such that tunneling out of the largedot is negligible for the time scales of interest. TheCoulomb term is given by V c x 1 ,x 2 =q 2 /4 r o r, with r=x 1 −x 2 2 + c 2 and r =13 the relative permittivity in GaAs.This simplified form of the Coulomb interaction assumesthat all excitations take place in the x direction, whereas inthe y, z directions the electrons occupy at all times the lowesttransverse modes. To satisfy this assumption we choose forall the calculations c =20 nm, which gives a physicalconfinement length in the transverse y, z directions muchsmaller than that in x. The two-electron time-independentSchrödinger equation is solved numerically by the configurationinteraction method. The low-energy eigenstates consistof two closely spaced singlets and two triplets with a somewhatlarger energy gap to higher-lying states. Figure 1b11098-0121/2007/758/0853026085302-1©2007 The American Physical Society

GIAVARAS et al.FIG. 1. Color online a Quantum dot confining potential andeffective charge density in arbitrary units of the lowest tripleteigenstate. b Contour plot of the lowest triplet eigenstate. c Thetwo-lowest singlet and triplet pairs.shows the lowest triplet eigenstate o T x 1 ,x 2 antisymmetricand Fig. 1c shows typical energy levels. The correspondingeffective charge density of o T x 1 ,x 2 which is derivedby integrating the two-electron charge density over thePHYSICAL REVIEW B 75, 085302 2007spatial coordinates of one of the two electrons, i.e., T x=2q T o x,x 2 dx is shown in Fig. 1a. For this distributionone electron is bound in the small dot, whereas thesecond electron is at the left and right corners with equalprobability due to the symmetric potential. The charge densitiesassociated with all states in the low-lying manifold arein fact virtually identical to that shown in Fig. 1a.This can be understood from a simple Heitler-London pictureusing one-electron states derived from a Hartreeapproximation in which we replace the trapped electron inthe small dot with an effective Hartree potentialV H x=V c x,x c x 2 dx, where c x describes thesingle-electron state of the small dot. The second electronfeels the approximate potential V H x+V d x, which for thelowest two-electron states of interest acts as a double-wellpotential. The two lowest single-electron states, − x, + xbonding-antibonding, of this potential have a small energysplitting and peak in the left and right corners. Using theone-electron Hartree states we may form the two lowest electronstates independent of spin o x 1 ,x 2 = − x 1 c x 2 and 1 x 1 ,x 2 = + x 1 c x 2 and similarly for the nexthigher pair. Within the Hartree model this means that notonly the lowest singlet, triplet eigenstates have a distributionof the form of Fig. 1a, but also the first excited eigenstateswhich indeed agrees with the results that we derive from theexact diagonalization. Note that even though the Hartreemodel gives great insight into the two-electron energies anddistributions of the low-lying states, with remarkably littleeffort, the symmetric and antisymmetric combinations S,A x 1 ,x 2 = o x 1 ,x 2 ± o x 2 ,x 1 / 2 give a poor approximationto the singlet-triplet splitting, with even the incorrectsign. This is little improved by using more accurateone-electron states, e.g., from Hartree-Fock or density functionaltheory DFT, being a consequence of strong correlationsrequiring accurate solutions of the two-electron problem.However, as we discuss later, the system may bedescribed accurately by an extended Hubbard model.We point out that the restriction to a 1D system is mainlyfor simplicity of presentation and a 2D structure, with a numberof channels in the transverse direction, would be morerealistic for a device based on a GaAs heterostructure. However,it is straightforward to generalize the 1D results to 2Dfor rectangular dots with high aspect ratio for which the Coulombinteraction again leads to peaks in probability densitynear the opposite boundaries of the dots and a pair of isolatedsinglets and triplets, as has been shown in Ref. 32 for arelated two-electron system in a square confining potential.For the dynamics that we describe below it is important tointroduce left and right states which are formed by combiningthe two lowest eigenstates for singlet S L = S o + S S1 , R= S o − S 1 and triplet T L =− T o + T 1 , T R = T o + T 1 unnormalized.For these states one electron is bound in the small dotwhereas the second electron is localized to the correspondingcorner as we demonstrate in Fig. 2 for the triplet.Although the effective charge density of the low-lyingsinglets and triplets is virtually identical, the wave functionsare quite different and give rise to effective interactions betweenspins, responsible for their entanglement. To demonstratehow these states can generate spin entanglement and085302-2

SINGLET-TRIPLET FILTERING AND ENTANGLEMENT IN…PHYSICAL REVIEW B 75, 085302 2007FIG. 2. Color online Effective charge density for a left andb right triplet states. The quantum dot confining potential is alsoshown.give rise to singlet-triplet filtering, we first prepare the systemsuch that one electron is located on the left with spin upand the other in the central dot with spin down. A suitablenonentangled state of this type may be written as a superpositionof the left singlet and the left S z =0 triplet state,i.e., ↑↓ = S L S ↑↓ + T L T ↑↓ / 2, with the spin eigenstates S/T ↑↓ 1,2= ↑ 1 ↓ 2 ↓ 1 ↑ 2/ 2. An approximationto this state may be obtained in principle by applying a smallsource-drain bias V sd across the quantum dot structure toseparate the electrons with negligible overlap. The spins maythen be initialized using magnetic fields and a microwavepulse, as suggested for scalable qubit arrays. 7,14 If V sd isremoved, then oscillations in a similar fashion to a doubledot system and entanglement will develop with a typicaltime for the triplet component of the wave function on theorder of /E T 1 s, where E T =E T 1 −E T o is the energysplitting of the two lowest triplet eigenstates when V sd =0similarly for the singlet. To speed up the process we canapply a time-dependent gate voltage that will tune in such away the dot potential so as to increase this energy splitting.In this work we have modelled the gate voltage potentialusing the expression V g x,t=−V p texp−x 2 /2l 2 g l g130 nm, which is driven by a triangular pulse of the formV p t =2V bT pt, 0 t T p /2− 2V bT pt +2V b , T p /2 t T p ,2FIG. 3. Effective charge density for singlet and triplet componentsat the final time for a a case in the filtering regime and bthe most general case.where T p is the period of the pulse and V b is the maximumgate voltage. We have studied the case for which the gatingrate =dV p /dt is such that to a good approximation onlythe two lowest singlet and two lowest triplet states are involvedin the dynamics. During the first half of the cycle0tT p /2 the effect of the voltage is to decrease theeffective width of the dot thereby increasing the energy splittingand the interaction, whereas during the second half ofthe cycle T p /2tT p the process is reversed. Investigationof the optimum pulse shape and possible nonadiabaticeffects are beyond the scope of this work.The dynamics of the two electrons is governed by thetime-dependent Schrödinger equation with the Hamiltonian1 and for a total potential V d x+V g x,t which is the sumof the dot confining potential and the time-dependent gatepotential. For t0, the spin eigenstates are unchanged forsinglet or triplet components, whereas the evolution of thecorresponding orbital states is given directly by the solutionof the time-dependent Schrödinger equation which is implementednumerically using the staggered-time algorithm proposedby Visscher. 33 The time evolution of the initial state ↑↓ is then determined by adding the separately determinedsinglet and triplet components. In Fig. 3 we show two examplesof the final electron distribution for singlet and tripletcomponents. Specifically, in Fig. 3a the singlet correspondsto a good approximation to the right state, whereas the tripletcorresponds to the left state. This is the ideal filtering regimetogether with the opposite limit in which at the final timethe two components occupy different spatial regions. Notethat in this regime the two electrons are fully entangled providedthe measurement domain is restricted either to the leftor right region, detecting the singlet or the S z =0 triplet state,respectively, which are fully entangled states. Performing anadditional measurement but initializing the electron spins toa purely triplet state either of S z =±1 reveals the occupationregion of the triplet state hence offering a way to discriminatebetween singlet and triplet components. Note that in themost general case which is shown in Fig. 3b the final twoelectronstate is a superposition of left and right states forboth singlet and triplet components. Clearly, a measurementwith restriction to either the left domain or the right domainwould probe a partially entangled state.To understand the dynamics and the difference betweensinglet and triplet components we can write the time dependence,for example of the singlet, using the correspondingtwo lowest energy states at all times, i.e., S=C S L S L +C S R S R in the left-right basis with the amplitudesC S L =e −iS cos S and C S R =ie −iS sin S . The parameters S085302-3

GIAVARAS et al.PHYSICAL REVIEW B 75, 085302 2007C R = 2P nsfP sf 1/2, 4P sf + P nsfwith P nsf =C nsf 2 and P sf =C sf 2 , thusC R = P R S + P R T 2 −4P R S P R T cos 2 1/2P R S + P RT, 5FIG. 4. Energy splitting of the two lowest levels for singlet andtriplet components, as a function of time for one cycle in the filteringregime.= t 0 E S 1 t+E S o tdt/2 and S = t 0 E S 1 t−E S o tdt/2depend on the two lowest energy levels and change withtime. Note that S determines the amount of filtering becauseit determines the period of the oscillations between left andright states, whereas both S and S determine the degree ofentanglement as we demonstrate below. In Fig. 4 we showthe energy splitting of the two lowest levels the energies arecalculated from instantaneous solutions as a function oftime for one cycle in the filtering regime which gives fullyentangled electrons in time T p 0.3 ns for V b 2.8 meV.We see that the splitting is larger for the singlet and thismeans that S T , i.e., the singlet component oscillatesfaster than the triplet. We can understand this effect with aHartree approximation: because the small dot has only onebound energy level c , the tunneling time the time that theelectron in the left corner needs in order to move to the rightcorner for the triplet is longer due to the Pauli blocking.Within the Hartree model the electron in the large dot needsto tunnel through an effective double barrier due to the Coulombrepulsion from the trapped electron in the small dot,which has only one resonance energy level c +U c with U cthe Coulomb energy when both electrons occupy c andcorresponds to a singlet state. For a triplet resonance to existthe small dot needs to have at least two bound levels due tothe Pauli principle. It is worth noting that the efficiency ofthis singlet-triplet filtering mechanism and the induced entanglementhas been also demonstrated in scattering problemsbetween a static and a flying qubit. 25,26To quantify the degree of entanglement which arises as aconsequence of the interaction when the electrons are closetogether we have calculated concurrence 34 at the final time,from spin-flip, P sf , and non-spin-flip, P nsf , probabilities. Forexample, at the right region the amplitudes for these probabilitiesareC nsf = C R T S+ C R, C sf = C R T S− C R, 322which are derived by writing the corresponding final state ↑↓ t f in the basis of spin eigenstates. Concurrence at theright region is then given bywhere we have set P S R =C S R 2 =sin 2 S and P T R =C T R 2=sin 2 T and the relative phase = S − T . A similar expressioncan be derived for the left and even for the total region.Note that by definition 0C R 1 where the limit C R =0 correspondsto unentangled electrons, whereas the limit C R =1to fully entangled electrons. The measurement in the rightregion is meaningful only when P S R 0, and/or P T R 0, i.e.,when the right state is occupied for singlet and/or triplet. Wesee from Eq. 4 that C R =1 when P sf = P nsf or equivalentlyfrom Eq. 5 when cos =0 and/or when P S R =0 and simultaneouslyP T R 0 or vice versa.In Fig. 5 we present the dependence of probabilities,phase shift, and concurrence as a function of maximum gatevoltage V b for a fixed gating rate 18 meV/ns which ensuresthat to a good approximation only the two lowesteigenstates for both singlet and triplet components are involvedinto the dynamics. Figure 5a shows the probabilitiesP S R , P T R and the absolute magnitude of the quantity cos calculatedat the final time vs V b . We see that the probabilitiesP S TR , P R do not oscillate for small gates voltages V b2.5 meV but rather they increase because the energysplitting of the two lowest levels in this limit remains relativelysmall and therefore the tunneling time is long. Furtherincrease of the gate voltage induces well-defined oscillations.In particular the probability of the singlet component oscillatesfaster than the triplet because the energy splitting whichdetermines the frequency of the oscillations is larger for thesinglet. Note also that the frequency of oscillations increaseswith gate voltage for both components following the increaseof the energy splitting. Figure 5b shows the concurrence C Rand the spin-flip P sf and non-spin-flip P nsf probabilities calculatedat the final time. When these probabilities are equalthe entanglement is maximum whereas when one of them iszero the electrons remain unentangled. For a gate voltageV b 2.8 meV we have an example of the ideal filtering regimefor which P S R 0, P T R 1, and P sf = P nsf giving C R =1,since only a triplet state occupies the right region. The genericcondition for ideal filtering is when sin S =1 and simultaneouslysin T =0 or vice versa. Other filtering casesnonideal which give C R =1 occur when one of the componentsis zero, say P S R =0 and the other is nonzero P T R 0, butnot, however, equal to 1. Finally an interesting limit is whenP S R = P T R for which the concurrence reduces to the simple expressionC R =sin which depends only on the relativephase between singlet and triplet components.Figure 6 presents the dependence of various quantities asa function of the pulse duration time for two different gatevoltages. As we see by comparing Figs. 6a and 6b theperiod of the oscillations can be controlled with the value ofthe maximum gate voltage with the larger V b inducing fasteroscillations due to the larger energy splitting.085302-4

SINGLET-TRIPLET FILTERING AND ENTANGLEMENT IN…PHYSICAL REVIEW B 75, 085302 2007FIG. 5. Color online Dependence at the finaltime, as a function of maximum gate voltagefor fixed gating rate, of a absolute value ofcos and right probabilities for singlet and tripletcomponents, and b concurrence, spin-flip, andnon-spin-flip probabilities calculated for the rightregion.Finally, we point out that the low-lying spin states relevantto the dynamics of this entangler which involve virtuallyidentical charge distributions at the left, center, and rightof the dot structure suggest that this charge-spin system maybe modelled by a three-site Hubbard model of the formH eff = n L + n R + c n C + C † i, C C, + H.c.,i=L,R+ Un L,↑ n L,↓ + n R,↑ n R,↓ + U c n C,↑ n C,↓ + Vn L n C + n C n R ,6where n i = n i, = C † i, C i, , i=L,R,C, and C † i, C i, createsdestroys an electron on site i with spin . The on-siteorbital energy for the left and right sites is and for thecentral site is c . The on-site Coulomb energy is denoted byU for the left and right sites and by U c for the central site.The nearest site Coulomb energy is V and finally expressesthe hopping between nearest sites. Note that all the physicalparameters , c ,U,U c ,V, may be estimated from theHartree approximation described earlier. Working within therestricted subspace for which the two-electron basis statesconsist of six singlets and three triplets, we can extract thecorrect energy splitting of the two lowest eigenstates forboth singlet and triplet, the corresponding eigenstates andthe antiferromagnetic exchange energy. Solution of this timedependentmodel Eq. 6 does indeed show qualitatively thesame behavior as the original continuous problem but withconsiderable saving in computer time once the timedependentparameters are known. The analysis based on theeffective Hamiltonian Eq. 6 indicates that the physical behaviorthat we have demonstrated can also be realized in atriple quantum dot structure. More importantly, the Hubbardmodel gives insight into the behavior of the system since it isreadily mapped onto an effective charge-spin model for thelow-lying manifold of two singlets and two triplets. In particular,for singlets there are two processes by which an electronmay tunnel from left to right, as shown in Fig. 7. Theprocess a, that is only allowed for singlets, has amplitudeFIG. 6. Color online Dependence at the finaltime of quantities in Fig. 5 as a function ofpulse duration time for two different maximumgate voltages. a V b =2.75 meV and b V b=5 meV.085302-5

GIAVARAS et al.FIG. 7. Illustration of the two-electron spin dynamics in a threestepprocess, I, II, III. The process a is allowed only for singletswhereas the process b is allowed for both singlets and triplets. 1 = 2 /U−V+ c −J where J is the Heisenberg exchangeenergy between an electron spin on the central siteand one on either the left or the right site. On the other hand,process b is valid for both singlets and triplets and hasPHYSICAL REVIEW B 75, 085302 2007amplitude 2 = 2 /− c −V. The relative rate of tunnelingfor singlet and triplet is thus tuned by the gate by changing c , since making c more negative the energy denominator in 1 decreases whereas that of 2 increases.In summary we have presented a quantum dot structureand described the two-electron distribution for the lowestsinglet and triplet states. By studying the electron dynamicsdue to a time-dependent gate voltage we showed how we caninduce a singlet-triplet filtering based on the Pauli blockingeffect for the triplet state. This can generate full spin entanglementwithin the order of 0.3 ns. Both the degree offiltering and entanglement can be efficiently controlled withthe gating rate and the maximum applied gate voltage.G.G. thanks UK EPSRC for funding. This work was supportedby the UK Ministry of Defence.1 L. P. Kouwenhoven, T. H. Oosterkamp, M. W. S. Danoeastro, M.Eto, D. G. Austing, T. Honda, and S. Tarucha, Science 278,1788 1997.2 S. Tarucha, D. G. Austing, T. Honda, R. J. van der Hage, and L.P. Kouwenhoven, Phys. Rev. Lett. 77, 3613 1996.3 L. P. Kouwenhoven, D. G. Austing, and S. Tarucha, Rep. Prog.Phys. 64, 701 2001.4 W. G. van der Wiel, S. De Franceschi, J. M. Elzerman, T.Fujisawa, S. Tarucha, and L. P. Kouwenhoven, Rev. Mod. Phys.75, 12003.5 D. H. Cobden and J. Nygard, Phys. Rev. Lett. 89, 046803 2002.6 S. Moriyama, T. Fuse, M. Suzuki, Y. Aoyagi, and K. Ishibashi,Phys. Rev. Lett. 94, 186806 2005.7 D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 1998.8 L. Fedichkin, M. Yanchenko, and K. A. Valiev, Nanotechnology11, 387 2000.9 G. Burkard, D. Loss, and D. P. DiVincenzo, Phys. Rev. B 59,2070 1999.10 J. Schliemann, D. Loss, and A. H. MacDonald, Phys. Rev. B 63,085311 2001.11 X. Hu and S. Das. Sarma, Phys. Rev. A 61, 062301 2000.12 M. Friesen, P. Rugheimer, D. E. Savage, M. G. Lagally, D. W.van der Weide, R. Joynt, and M. A. Eriksson, Phys. Rev. B 67,121301R 2003.13 P. Zhang, Q. K. Xue, X. G. Zhao, and X. C. Xie, Phys. Rev. A 66,022117 2002.14 F. H. L. Koppens, C. Buizert, K. J. Tielrooij, I. T. Vink, K. C.Nowack, T. Meunier, L. P. Kouwenhoven, and L. M. K. Vandersypen,Nature London 442, 766 2006.15 R. Hanson, L. H. Willems van Beveren, I. T. Vink, J. M. Elzerman,W. J. M. Naber, F. H. L. Koppens, L. P. Kouwenhoven, andL. M. K. Vandersypen, Phys. Rev. Lett. 94, 196802 2005.16 T. Hayashi, T. Fujisawa, H. D. Cheong, Y. H. Jeong, and Y.Hirayama, Phys. Rev. Lett. 91, 226804 2003.17 J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby,M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard,Science 309, 2180 2005.18 G. Burkard, D. Loss, and E. V. Sukhorukov, Phys. Rev. B 61,R16303 2000.19 P. Recher, E. V. Sukhorukov, and D. Loss, Phys. Rev. B 63,165314 2001.20 D. S. Saraga, B. L. Altshuler, D. Loss, and R. M. Westervelt,Phys. Rev. B 71, 045338 2005.21 D. S. Saraga, B. L. Altshuler, D. Loss, and R. M. Westervelt,Phys. Rev. Lett. 92, 246803 2004.22 C. H. W. Barnes, J. M. Shilton, and A. M. Robinson, Phys. Rev.B 62, 8410 2000.23 G. Gumbs and Y. Abranyos, Phys. Rev. A 70, 050302R 2004.24 S. Furuta, C. H. W. Barnes, and C. J. L. Doran, Phys. Rev. B 70,205320 2004.25 G. Giavaras, J. H. Jefferson, A. Ramšak, T. P. Spiller, and C. J.Lambert, Phys. Rev. B 74, 195341 2006.26 J. H. Jefferson, A. Ramšak, and T. Rejec, Europhys. Lett. 75, 7642006.27 R. Hanson and G. Burkard, cond-mat/0605576 unpublished.28 J. M. Taylor, H.-A. Engel, W. Dür, A. Yacoby, C. M. Marcus, P.Zoller, and M. D. Lukin, Nat. Phys. 1, 177 2005.29 G. W. Bryant, Phys. Rev. Lett. 59, 1140 1987.30 C. E. Creffield, W. Häusler, J. H. Jefferson, and S. Sarkar, Phys.Rev. B 59, 10719 1999.31 For two electrons x 1 , 1 ,x 2 , 2 =x 1 ,x 2 1 , 2 with symmetric antisymmetric for singlet triplet under electronexchange. Because the Hamiltonian does not contain any spindependent term we can consider only the spatial components.32 J. H. Jefferson, M. Fearn, D. L. J. Tipton, and T. P. Spiller, Phys.Rev. A 66, 042328 2002.33 P. B. Visscher, Comput. Phys. 5, 596 1991.34 W. K. Wootters, Phys. Rev. Lett. 80, 2245 1998.085302-6